Inventing a New Model

1. Inventing a New Model Yesterday, you used the number line to help you think about adding integers. In this problem, you will explore another way to model the addition of integers. Amber’s mother is an accountant. One day, Amber heard her mother talking to a client on the phone. During the conversation, her mother used the phrases “in the red” and “in the black”.

2. That evening at dinner, Amber asked her mother what these terms meant. Her mother said: “When people in business talk about income and expenses, they often use colors to often describe the numbers they are dealing with. Black refers to profits (or income); A company that is making money, or has money, is “in the black”; Red refers to losses (or expenses). A company that is losing money or owes money, is “in the red.”

3. Amber was studying Integers in her math class and thought she could use these ideas of “in theblack” and “in the red” to model the addition of positive and negative integers. Chip Board = +1 = -1 Her model uses a chip board and black and red chips. Each black chip represents +1 and each red chip represents -1. For example, this chip board shows a value of +5:

4. = +1 = -1 Chip Board This chip board shows a value of -5

5. Chip Board To represent -4 + -3 =, Amber started with an empty chip board. She represented -4 by putting four red chips on the board. To represent the addition of -3, she put three more red chips on the board. Since there were 7 red chips on the board, Amber concluded that the sum (answer) of -4 and -3 is -7. -4 + -3 = -7

6. = +1 = -1 Chip Board Amber showed her idea to Adil. Adil liked Amber’s model, but he wasn’t sure how to use it to add a negative number and a positive integer. Amber modeled the next problem using the chip board: -4 + +5 Amber started with an empty chip board. She put 4 red chips on the board to represent -4. To add +5, Amber added five black chips to the board.

7. Amber said that next she had to simplify the board so that the answer would be easier to read. She reminded Adil that +1 and -1 are opposites, they add to 0. So a pair of one black chip (+1) and one red chip (-1) represents 0. Amber circled as many black-red pairs as she could. Chip Board -4 + +5 = +1 Since each pair represents 0, the pairs are removed from the chip board and not counted. Only one black chip remains, that represents the answer or sum. +1

8. Chip Board Now, let’s try one more addition problem together. +8 + -6 = +2 First you put 8 black chips to represent +8. Then you add 7 red chips to represent -6. The answer to the sum is +2 because only 2 black chips are left on the chip board. Remove all the black-red pairs because they represent 0.